Elliptic Operators of Arbitrary Order

2018 ◽  
pp. 83-113
Author(s):  
David E. Edmunds ◽  
W. Desmond Evans
Keyword(s):  
Arbitrary Order ◽  
Elliptic Operators

Continuity Points Via Riesz Potentials for ℂ-Elliptic Operators

2020 ◽  
Vol 71 (4) ◽  
pp. 1201-1218
Author(s):  
Lars Diening ◽  
Franz Gmeineder
Keyword(s):  
Differential Operator ◽  
Arbitrary Order ◽  
Riesz Potential ◽  
Elliptic Operators ◽  
Gradient Estimates ◽  
Elliptic Differential Operator ◽  
Riesz Potentials

Abstract We establish a Riesz potential criterion for Lebesgue continuity points of functions of bounded $\mathbb{A}$-variation, where $\mathbb{A}$ is a $\mathbb{C}$-elliptic differential operator of arbitrary order. This result generalizes a potential criterion that is known for full gradients to the case where full gradient estimates are not available by virtue of Ornstein’s non-inequality.

A computer-assisted proof method of the invertibility to elliptic operators

2014 ◽  
Vol 1 ◽  
pp. 816-819
Author(s):  
Akitoshi Takayasu ◽  
Shin'ichi Oishi
Keyword(s):  
Elliptic Operators ◽  
Computer Assisted ◽  
Computer Assisted Proof

Degenerate Diffusion Operators Arising in Population Biology (AM-185)

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Adjoint Operator ◽  
Integral Kernel ◽  
Mathematical Finance ◽  
Elliptic Operators ◽  
Complete Analysis ◽  
Time Behavior ◽  
Degenerate Elliptic Operators ◽  
Regularity Properties ◽  
Degenerate Elliptic ◽  
Mathematical Foundations

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the martingale problem and therefore the existence of the associated Markov process. The book uses an “integral kernel method” to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. The book establishes the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. It shows that the semigroups defined by these operators have holomorphic extensions to the right half plane. The book also demonstrates precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

scholarly journals Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝN

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Anup Biswas ◽  
Prasun Roychowdhury
Keyword(s):  
Eigenvalue Problem ◽  
Principal Eigenvalue ◽  
Unbounded Domains ◽  
Elliptic Operators ◽  
Maximum Principles ◽  
Generalized Eigenvalues ◽  
Nonlinear Elliptic ◽  
Generalized Eigenvalue ◽  
General Convex ◽  
Appl Math

AbstractWe study the generalized eigenvalue problem in {\mathbb{R}^{N}} for a general convex nonlinear elliptic operator which is locally elliptic and positively 1-homogeneous. Generalizing [H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math. 68 2015, 6, 1014–1065], we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.

scholarly journals The determinant of one-dimensional polyharmonic operators of arbitrary order

2020 ◽  
Vol 279 (12) ◽  
pp. 108783
Author(s):  
Pedro Freitas ◽  
Jiří Lipovský
Keyword(s):  
Arbitrary Order ◽  
One Dimensional
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